论文标题
多-D Poisson-Nernst-Planck系统的有效，积极和能量稳定方案
Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems
论文作者
论文摘要
在本文中，我们设计，分析和数值验证了正面和能量消散方案，以求解Poisson-Nernst-Planck（PNP）方程的时间依赖性多维系统，该系统在生物膜通道和半导体式的生物膜通道的建模中有很大使用。基于系统重新制定的半平整时间离散化提供了一个良好的椭圆系统，该系统被证明可以为任意时间步骤保留解决方案的阳性。表明，第一阶（及时）完全消失的方案无条件地保留溶液的阳性和质量保护，并且只有轻度$ O（1）$（1）$时间步长限制。该方案还证明可以保留稳态。对于具有较大时间步长的完全二阶（在时间和空间中）方案，解决方案阳性由局部规模限制器恢复，该局部比例限制器可维持空间精度。这些方案易于实施。几个三维数字示例验证了我们的理论发现，并证明了所提出的方案的准确性，效率和鲁棒性以及对稳态状态的快速方法。
In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck (PNP) equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild $O(1)$ time step restriction. The scheme is also shown to preserve the steady-state. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady states.
